- Level: International Baccalaureate
- Subject: Maths
- Word count: 1683
Continued Fractions
Extracts from this document...
Introduction
Continued Fractions
A continued fraction is any mathematical expression in the form of:
Where a0 is always and integer, and all other ‘’s such as a1, a2, and a3 are positive integers. The number of terms can either be finite or infinite. A more convenient way to denote continued fractions such as the one above would be to denote it by:
Finite Continued Fractions
A finite continued fraction is an expression such as the one shown above which could end. Every rational number can be equated to a finite continued fraction. The only skill needed would be division of fractions.
Infinite Continued Fractions
Unlike the finite continued fractions, the chain of fractions never ends in an infinite continued fraction. Every irrational number can be equated to an infinite continued fraction. This fact was discovered and proven by the Swiss Mathematician, Leonhard Euler (1707-1783). Some of Euler’s infinite continued fractions are as we will see below:
A way to summarise this expression is to let denote the value of the continued fraction.
Usage of Continued Fractions
Continued fractions could be used to solve certain quadratic equations of the second degree. Solving a quadratic equation using the
Middle
The same trend is brought up by the graph of and . As the value of rises, will oscillate until reaches 25. Only then will the value of converge to a consistent value of 0. This value will stay the same as long as 25.
- When trying to determine the 200th term, we couldn’t obtain a negative value, only a positive value. When the value of is still below 25, it oscillates between values which sometimes range from negative to positive values, but then as the value of increases it will converge to a constant specific positive value. That means a negative value could only be obtained if
- According to the table in question two, the exact value for the continued fraction is 1.
Conclusion
Bibliography
http://images.google.com.sg/imgres?imgurl=http://upload.wikimedia.org/math/a/b/d/abda6e157fb26b5fb84d43df36a8ad31.png&imgrefurl=http://en.wikipedia.org/wiki/Solving_quadratic_equations_with_continued_fractions&h=169&w=296&sz=2&hl=en&start=14&um=1&usg=__bg1UGcYe_dOsasg8Zdzt-Vawfjk=&tbnid=qfG0CexZoK9QDM:&tbnh=66&tbnw=116&prev=/images%3Fq%3Dcontinued%2Binfinite%2Bfraction%26um%3D1%26hl%3Den%26sa%3DG
http://images.google.com.sg/imgres?imgurl=http://www.jcu.edu/math/vignettes/v18im7.gif&imgrefurl=http://www.jcu.edu/math/vignettes/continued.htm&h=279&w=322&sz=3&hl=en&start=9&um=1&usg=__2U0RUn8JPqKD-Mom_NAD3Wge-M4=&tbnid=agACwiBwpt7vbM:&tbnh=102&tbnw=118&prev=/images%3Fq%3Dcontinued%2Bfinite%2Bfractions%26um%3D1%26hl%3Den%26sa%3DG
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